Integrand size = 29, antiderivative size = 880 \[ \int x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 \, dx=-\frac {37384 b^2 d^2 \sqrt {d-c^2 d x^2}}{694575 c^4}+\frac {3358 b^2 d^2 x^2 \sqrt {d-c^2 d x^2}}{694575 c^2}+\frac {484 b^2 d^2 x^4 \sqrt {d-c^2 d x^2}}{77175}-\frac {10 b^2 c^2 d^2 x^6 \sqrt {d-c^2 d x^2}}{3087}+\frac {4 a b d^2 x \sqrt {d-c^2 d x^2}}{63 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {16 b^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{2835 c^4 (1-c x) (1+c x)}+\frac {8 b^2 d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{8505 c^4 (1-c x) (1+c x)}+\frac {2 b^2 d^2 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2}}{4725 c^4 (1-c x) (1+c x)}-\frac {20 b^2 d^2 \left (1-c^2 x^2\right )^4 \sqrt {d-c^2 d x^2}}{3969 c^4 (1-c x) (1+c x)}+\frac {2 b^2 d^2 \left (1-c^2 x^2\right )^5 \sqrt {d-c^2 d x^2}}{729 c^4 (1-c x) (1+c x)}+\frac {4 b^2 d^2 x \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{63 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b d^2 x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{189 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c d^2 x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{21 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {38 b c^3 d^2 x^7 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{441 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c^5 d^2 x^9 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{81 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{63 c^4}-\frac {d^2 x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{63 c^2}+\frac {1}{21} d^2 x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2+\frac {5}{63} d x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2+\frac {1}{9} x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 \]
5/63*d*x^4*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))^2+1/9*x^4*(-c^2*d*x^2+d )^(5/2)*(a+b*arccosh(c*x))^2-37384/694575*b^2*d^2*(-c^2*d*x^2+d)^(1/2)/c^4 +3358/694575*b^2*d^2*x^2*(-c^2*d*x^2+d)^(1/2)/c^2+484/77175*b^2*d^2*x^4*(- c^2*d*x^2+d)^(1/2)-10/3087*b^2*c^2*d^2*x^6*(-c^2*d*x^2+d)^(1/2)+16/2835*b^ 2*d^2*(-c^2*x^2+1)*(-c^2*d*x^2+d)^(1/2)/c^4/(-c*x+1)/(c*x+1)+8/8505*b^2*d^ 2*(-c^2*x^2+1)^2*(-c^2*d*x^2+d)^(1/2)/c^4/(-c*x+1)/(c*x+1)+2/4725*b^2*d^2* (-c^2*x^2+1)^3*(-c^2*d*x^2+d)^(1/2)/c^4/(-c*x+1)/(c*x+1)-20/3969*b^2*d^2*( -c^2*x^2+1)^4*(-c^2*d*x^2+d)^(1/2)/c^4/(-c*x+1)/(c*x+1)+2/729*b^2*d^2*(-c^ 2*x^2+1)^5*(-c^2*d*x^2+d)^(1/2)/c^4/(-c*x+1)/(c*x+1)-2/63*d^2*(a+b*arccosh (c*x))^2*(-c^2*d*x^2+d)^(1/2)/c^4-1/63*d^2*x^2*(a+b*arccosh(c*x))^2*(-c^2* d*x^2+d)^(1/2)/c^2+1/21*d^2*x^4*(a+b*arccosh(c*x))^2*(-c^2*d*x^2+d)^(1/2)+ 4/63*a*b*d^2*x*(-c^2*d*x^2+d)^(1/2)/c^3/(c*x-1)^(1/2)/(c*x+1)^(1/2)+4/63*b ^2*d^2*x*arccosh(c*x)*(-c^2*d*x^2+d)^(1/2)/c^3/(c*x-1)^(1/2)/(c*x+1)^(1/2) +2/189*b*d^2*x^3*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/c/(c*x-1)^(1/2)/( c*x+1)^(1/2)-2/21*b*c*d^2*x^5*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/(c*x -1)^(1/2)/(c*x+1)^(1/2)+38/441*b*c^3*d^2*x^7*(a+b*arccosh(c*x))*(-c^2*d*x^ 2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-2/81*b*c^5*d^2*x^9*(a+b*arccosh(c*x ))*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)
Time = 0.60 (sec) , antiderivative size = 288, normalized size of antiderivative = 0.33 \[ \int x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 \, dx=\frac {d^2 \sqrt {d-c^2 d x^2} \left (3969 a^2 \left (-1+c^2 x^2\right )^4 \left (2+7 c^2 x^2\right )-126 a b c x \sqrt {-1+c x} \sqrt {1+c x} \left (-126-21 c^2 x^2+189 c^4 x^4-171 c^6 x^6+49 c^8 x^8\right )+2 b^2 \left (6140-7039 c^2 x^2-106 c^4 x^4+2152 c^6 x^6-1490 c^8 x^8+343 c^{10} x^{10}\right )+126 b \left (63 a \left (-1+c^2 x^2\right )^4 \left (2+7 c^2 x^2\right )+b c x \sqrt {-1+c x} \sqrt {1+c x} \left (126+21 c^2 x^2-189 c^4 x^4+171 c^6 x^6-49 c^8 x^8\right )\right ) \text {arccosh}(c x)+3969 b^2 \left (-1+c^2 x^2\right )^4 \left (2+7 c^2 x^2\right ) \text {arccosh}(c x)^2\right )}{250047 c^4 \left (-1+c^2 x^2\right )} \]
(d^2*Sqrt[d - c^2*d*x^2]*(3969*a^2*(-1 + c^2*x^2)^4*(2 + 7*c^2*x^2) - 126* a*b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-126 - 21*c^2*x^2 + 189*c^4*x^4 - 17 1*c^6*x^6 + 49*c^8*x^8) + 2*b^2*(6140 - 7039*c^2*x^2 - 106*c^4*x^4 + 2152* c^6*x^6 - 1490*c^8*x^8 + 343*c^10*x^10) + 126*b*(63*a*(-1 + c^2*x^2)^4*(2 + 7*c^2*x^2) + b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(126 + 21*c^2*x^2 - 189* c^4*x^4 + 171*c^6*x^6 - 49*c^8*x^8))*ArcCosh[c*x] + 3969*b^2*(-1 + c^2*x^2 )^4*(2 + 7*c^2*x^2)*ArcCosh[c*x]^2))/(250047*c^4*(-1 + c^2*x^2))
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 \, dx\) |
| \(\Big \downarrow \) 6345 |
| \(\displaystyle -\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \int x^4 (1-c x)^2 (c x+1)^2 (a+b \text {arccosh}(c x))dx}{9 \sqrt {c x-1} \sqrt {c x+1}}+\frac {5}{9} d \int x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2dx+\frac {1}{9} x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 6327 |
| \(\displaystyle -\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \int x^4 \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))dx}{9 \sqrt {c x-1} \sqrt {c x+1}}+\frac {5}{9} d \int x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2dx+\frac {1}{9} x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 6336 |
| \(\displaystyle \frac {5}{9} d \int x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2dx-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (-b c \int \frac {x^5 \left (35 c^4 x^4-90 c^2 x^2+63\right )}{315 \sqrt {c x-1} \sqrt {c x+1}}dx+\frac {1}{9} c^4 x^9 (a+b \text {arccosh}(c x))-\frac {2}{7} c^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{5} x^5 (a+b \text {arccosh}(c x))\right )}{9 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{9} x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{9} d \int x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2dx-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (-\frac {1}{315} b c \int \frac {x^5 \left (35 c^4 x^4-90 c^2 x^2+63\right )}{\sqrt {c x-1} \sqrt {c x+1}}dx+\frac {1}{9} c^4 x^9 (a+b \text {arccosh}(c x))-\frac {2}{7} c^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{5} x^5 (a+b \text {arccosh}(c x))\right )}{9 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{9} x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 1905 |
| \(\displaystyle \frac {5}{9} d \int x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2dx-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (-\frac {b c \sqrt {c^2 x^2-1} \int \frac {x^5 \left (35 c^4 x^4-90 c^2 x^2+63\right )}{\sqrt {c^2 x^2-1}}dx}{315 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{9} c^4 x^9 (a+b \text {arccosh}(c x))-\frac {2}{7} c^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{5} x^5 (a+b \text {arccosh}(c x))\right )}{9 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{9} x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 1578 |
| \(\displaystyle \frac {5}{9} d \int x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2dx-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (-\frac {b c \sqrt {c^2 x^2-1} \int \frac {x^4 \left (35 c^4 x^4-90 c^2 x^2+63\right )}{\sqrt {c^2 x^2-1}}dx^2}{630 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{9} c^4 x^9 (a+b \text {arccosh}(c x))-\frac {2}{7} c^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{5} x^5 (a+b \text {arccosh}(c x))\right )}{9 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{9} x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 1195 |
| \(\displaystyle \frac {5}{9} d \int x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2dx-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (-\frac {b c \sqrt {c^2 x^2-1} \int \left (\frac {35 \left (c^2 x^2-1\right )^{7/2}}{c^4}+\frac {50 \left (c^2 x^2-1\right )^{5/2}}{c^4}+\frac {3 \left (c^2 x^2-1\right )^{3/2}}{c^4}-\frac {4 \sqrt {c^2 x^2-1}}{c^4}+\frac {8}{c^4 \sqrt {c^2 x^2-1}}\right )dx^2}{630 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{9} c^4 x^9 (a+b \text {arccosh}(c x))-\frac {2}{7} c^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{5} x^5 (a+b \text {arccosh}(c x))\right )}{9 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{9} x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {5}{9} d \int x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2dx+\frac {1}{9} x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{9} c^4 x^9 (a+b \text {arccosh}(c x))-\frac {2}{7} c^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{5} x^5 (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {70 \left (c^2 x^2-1\right )^{9/2}}{9 c^6}+\frac {100 \left (c^2 x^2-1\right )^{7/2}}{7 c^6}+\frac {6 \left (c^2 x^2-1\right )^{5/2}}{5 c^6}-\frac {8 \left (c^2 x^2-1\right )^{3/2}}{3 c^6}+\frac {16 \sqrt {c^2 x^2-1}}{c^6}\right )}{630 \sqrt {c x-1} \sqrt {c x+1}}\right )}{9 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 6345 |
| \(\displaystyle \frac {5}{9} d \left (\frac {2 b c d \sqrt {d-c^2 d x^2} \int -x^4 (1-c x) (c x+1) (a+b \text {arccosh}(c x))dx}{7 \sqrt {c x-1} \sqrt {c x+1}}+\frac {3}{7} d \int x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2dx+\frac {1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2\right )+\frac {1}{9} x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{9} c^4 x^9 (a+b \text {arccosh}(c x))-\frac {2}{7} c^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{5} x^5 (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {70 \left (c^2 x^2-1\right )^{9/2}}{9 c^6}+\frac {100 \left (c^2 x^2-1\right )^{7/2}}{7 c^6}+\frac {6 \left (c^2 x^2-1\right )^{5/2}}{5 c^6}-\frac {8 \left (c^2 x^2-1\right )^{3/2}}{3 c^6}+\frac {16 \sqrt {c^2 x^2-1}}{c^6}\right )}{630 \sqrt {c x-1} \sqrt {c x+1}}\right )}{9 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {5}{9} d \left (-\frac {2 b c d \sqrt {d-c^2 d x^2} \int x^4 (1-c x) (c x+1) (a+b \text {arccosh}(c x))dx}{7 \sqrt {c x-1} \sqrt {c x+1}}+\frac {3}{7} d \int x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2dx+\frac {1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2\right )+\frac {1}{9} x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{9} c^4 x^9 (a+b \text {arccosh}(c x))-\frac {2}{7} c^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{5} x^5 (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {70 \left (c^2 x^2-1\right )^{9/2}}{9 c^6}+\frac {100 \left (c^2 x^2-1\right )^{7/2}}{7 c^6}+\frac {6 \left (c^2 x^2-1\right )^{5/2}}{5 c^6}-\frac {8 \left (c^2 x^2-1\right )^{3/2}}{3 c^6}+\frac {16 \sqrt {c^2 x^2-1}}{c^6}\right )}{630 \sqrt {c x-1} \sqrt {c x+1}}\right )}{9 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 6327 |
| \(\displaystyle \frac {5}{9} d \left (-\frac {2 b c d \sqrt {d-c^2 d x^2} \int x^4 \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))dx}{7 \sqrt {c x-1} \sqrt {c x+1}}+\frac {3}{7} d \int x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2dx+\frac {1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2\right )+\frac {1}{9} x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{9} c^4 x^9 (a+b \text {arccosh}(c x))-\frac {2}{7} c^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{5} x^5 (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {70 \left (c^2 x^2-1\right )^{9/2}}{9 c^6}+\frac {100 \left (c^2 x^2-1\right )^{7/2}}{7 c^6}+\frac {6 \left (c^2 x^2-1\right )^{5/2}}{5 c^6}-\frac {8 \left (c^2 x^2-1\right )^{3/2}}{3 c^6}+\frac {16 \sqrt {c^2 x^2-1}}{c^6}\right )}{630 \sqrt {c x-1} \sqrt {c x+1}}\right )}{9 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 6336 |
| \(\displaystyle \frac {5}{9} d \left (\frac {3}{7} d \int x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2dx-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-b c \int \frac {x^5 \left (7-5 c^2 x^2\right )}{35 \sqrt {c x-1} \sqrt {c x+1}}dx-\frac {1}{7} c^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{5} x^5 (a+b \text {arccosh}(c x))\right )}{7 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2\right )+\frac {1}{9} x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{9} c^4 x^9 (a+b \text {arccosh}(c x))-\frac {2}{7} c^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{5} x^5 (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {70 \left (c^2 x^2-1\right )^{9/2}}{9 c^6}+\frac {100 \left (c^2 x^2-1\right )^{7/2}}{7 c^6}+\frac {6 \left (c^2 x^2-1\right )^{5/2}}{5 c^6}-\frac {8 \left (c^2 x^2-1\right )^{3/2}}{3 c^6}+\frac {16 \sqrt {c^2 x^2-1}}{c^6}\right )}{630 \sqrt {c x-1} \sqrt {c x+1}}\right )}{9 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{9} d \left (\frac {3}{7} d \int x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2dx-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{35} b c \int \frac {x^5 \left (7-5 c^2 x^2\right )}{\sqrt {c x-1} \sqrt {c x+1}}dx-\frac {1}{7} c^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{5} x^5 (a+b \text {arccosh}(c x))\right )}{7 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2\right )+\frac {1}{9} x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{9} c^4 x^9 (a+b \text {arccosh}(c x))-\frac {2}{7} c^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{5} x^5 (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {70 \left (c^2 x^2-1\right )^{9/2}}{9 c^6}+\frac {100 \left (c^2 x^2-1\right )^{7/2}}{7 c^6}+\frac {6 \left (c^2 x^2-1\right )^{5/2}}{5 c^6}-\frac {8 \left (c^2 x^2-1\right )^{3/2}}{3 c^6}+\frac {16 \sqrt {c^2 x^2-1}}{c^6}\right )}{630 \sqrt {c x-1} \sqrt {c x+1}}\right )}{9 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 960 |
| \(\displaystyle \frac {5}{9} d \left (\frac {3}{7} d \int x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2dx-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{35} b c \left (\frac {19}{7} \int \frac {x^5}{\sqrt {c x-1} \sqrt {c x+1}}dx-\frac {5}{7} x^6 \sqrt {c x-1} \sqrt {c x+1}\right )-\frac {1}{7} c^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{5} x^5 (a+b \text {arccosh}(c x))\right )}{7 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2\right )+\frac {1}{9} x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{9} c^4 x^9 (a+b \text {arccosh}(c x))-\frac {2}{7} c^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{5} x^5 (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {70 \left (c^2 x^2-1\right )^{9/2}}{9 c^6}+\frac {100 \left (c^2 x^2-1\right )^{7/2}}{7 c^6}+\frac {6 \left (c^2 x^2-1\right )^{5/2}}{5 c^6}-\frac {8 \left (c^2 x^2-1\right )^{3/2}}{3 c^6}+\frac {16 \sqrt {c^2 x^2-1}}{c^6}\right )}{630 \sqrt {c x-1} \sqrt {c x+1}}\right )}{9 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 111 |
| \(\displaystyle \frac {5}{9} d \left (\frac {3}{7} d \int x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2dx-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{35} b c \left (\frac {19}{7} \left (\frac {\int \frac {4 x^3}{\sqrt {c x-1} \sqrt {c x+1}}dx}{5 c^2}+\frac {x^4 \sqrt {c x-1} \sqrt {c x+1}}{5 c^2}\right )-\frac {5}{7} x^6 \sqrt {c x-1} \sqrt {c x+1}\right )-\frac {1}{7} c^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{5} x^5 (a+b \text {arccosh}(c x))\right )}{7 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2\right )+\frac {1}{9} x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{9} c^4 x^9 (a+b \text {arccosh}(c x))-\frac {2}{7} c^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{5} x^5 (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {70 \left (c^2 x^2-1\right )^{9/2}}{9 c^6}+\frac {100 \left (c^2 x^2-1\right )^{7/2}}{7 c^6}+\frac {6 \left (c^2 x^2-1\right )^{5/2}}{5 c^6}-\frac {8 \left (c^2 x^2-1\right )^{3/2}}{3 c^6}+\frac {16 \sqrt {c^2 x^2-1}}{c^6}\right )}{630 \sqrt {c x-1} \sqrt {c x+1}}\right )}{9 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{9} d \left (\frac {3}{7} d \int x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2dx-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{35} b c \left (\frac {19}{7} \left (\frac {4 \int \frac {x^3}{\sqrt {c x-1} \sqrt {c x+1}}dx}{5 c^2}+\frac {x^4 \sqrt {c x-1} \sqrt {c x+1}}{5 c^2}\right )-\frac {5}{7} x^6 \sqrt {c x-1} \sqrt {c x+1}\right )-\frac {1}{7} c^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{5} x^5 (a+b \text {arccosh}(c x))\right )}{7 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2\right )+\frac {1}{9} x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{9} c^4 x^9 (a+b \text {arccosh}(c x))-\frac {2}{7} c^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{5} x^5 (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {70 \left (c^2 x^2-1\right )^{9/2}}{9 c^6}+\frac {100 \left (c^2 x^2-1\right )^{7/2}}{7 c^6}+\frac {6 \left (c^2 x^2-1\right )^{5/2}}{5 c^6}-\frac {8 \left (c^2 x^2-1\right )^{3/2}}{3 c^6}+\frac {16 \sqrt {c^2 x^2-1}}{c^6}\right )}{630 \sqrt {c x-1} \sqrt {c x+1}}\right )}{9 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 111 |
| \(\displaystyle \frac {5}{9} d \left (\frac {3}{7} d \int x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2dx-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{35} b c \left (\frac {19}{7} \left (\frac {4 \left (\frac {\int \frac {2 x}{\sqrt {c x-1} \sqrt {c x+1}}dx}{3 c^2}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}\right )}{5 c^2}+\frac {x^4 \sqrt {c x-1} \sqrt {c x+1}}{5 c^2}\right )-\frac {5}{7} x^6 \sqrt {c x-1} \sqrt {c x+1}\right )-\frac {1}{7} c^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{5} x^5 (a+b \text {arccosh}(c x))\right )}{7 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2\right )+\frac {1}{9} x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{9} c^4 x^9 (a+b \text {arccosh}(c x))-\frac {2}{7} c^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{5} x^5 (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {70 \left (c^2 x^2-1\right )^{9/2}}{9 c^6}+\frac {100 \left (c^2 x^2-1\right )^{7/2}}{7 c^6}+\frac {6 \left (c^2 x^2-1\right )^{5/2}}{5 c^6}-\frac {8 \left (c^2 x^2-1\right )^{3/2}}{3 c^6}+\frac {16 \sqrt {c^2 x^2-1}}{c^6}\right )}{630 \sqrt {c x-1} \sqrt {c x+1}}\right )}{9 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{9} d \left (\frac {3}{7} d \int x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2dx-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{35} b c \left (\frac {19}{7} \left (\frac {4 \left (\frac {2 \int \frac {x}{\sqrt {c x-1} \sqrt {c x+1}}dx}{3 c^2}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}\right )}{5 c^2}+\frac {x^4 \sqrt {c x-1} \sqrt {c x+1}}{5 c^2}\right )-\frac {5}{7} x^6 \sqrt {c x-1} \sqrt {c x+1}\right )-\frac {1}{7} c^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{5} x^5 (a+b \text {arccosh}(c x))\right )}{7 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2\right )+\frac {1}{9} x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{9} c^4 x^9 (a+b \text {arccosh}(c x))-\frac {2}{7} c^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{5} x^5 (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {70 \left (c^2 x^2-1\right )^{9/2}}{9 c^6}+\frac {100 \left (c^2 x^2-1\right )^{7/2}}{7 c^6}+\frac {6 \left (c^2 x^2-1\right )^{5/2}}{5 c^6}-\frac {8 \left (c^2 x^2-1\right )^{3/2}}{3 c^6}+\frac {16 \sqrt {c^2 x^2-1}}{c^6}\right )}{630 \sqrt {c x-1} \sqrt {c x+1}}\right )}{9 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 83 |
| \(\displaystyle \frac {5}{9} d \left (\frac {3}{7} d \int x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2dx+\frac {1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{7} c^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{5} x^5 (a+b \text {arccosh}(c x))-\frac {1}{35} b c \left (\frac {19}{7} \left (\frac {x^4 \sqrt {c x-1} \sqrt {c x+1}}{5 c^2}+\frac {4 \left (\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^4}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}\right )}{5 c^2}\right )-\frac {5}{7} x^6 \sqrt {c x-1} \sqrt {c x+1}\right )\right )}{7 \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{9} x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{9} c^4 x^9 (a+b \text {arccosh}(c x))-\frac {2}{7} c^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{5} x^5 (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {70 \left (c^2 x^2-1\right )^{9/2}}{9 c^6}+\frac {100 \left (c^2 x^2-1\right )^{7/2}}{7 c^6}+\frac {6 \left (c^2 x^2-1\right )^{5/2}}{5 c^6}-\frac {8 \left (c^2 x^2-1\right )^{3/2}}{3 c^6}+\frac {16 \sqrt {c^2 x^2-1}}{c^6}\right )}{630 \sqrt {c x-1} \sqrt {c x+1}}\right )}{9 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 6341 |
| \(\displaystyle \frac {5}{9} d \left (\frac {3}{7} d \left (-\frac {2 b c \sqrt {d-c^2 d x^2} \int x^4 (a+b \text {arccosh}(c x))dx}{5 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {d-c^2 d x^2} \int \frac {x^3 (a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{5 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{5} x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2\right )+\frac {1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{7} c^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{5} x^5 (a+b \text {arccosh}(c x))-\frac {1}{35} b c \left (\frac {19}{7} \left (\frac {x^4 \sqrt {c x-1} \sqrt {c x+1}}{5 c^2}+\frac {4 \left (\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^4}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}\right )}{5 c^2}\right )-\frac {5}{7} x^6 \sqrt {c x-1} \sqrt {c x+1}\right )\right )}{7 \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{9} x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{9} c^4 x^9 (a+b \text {arccosh}(c x))-\frac {2}{7} c^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{5} x^5 (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {70 \left (c^2 x^2-1\right )^{9/2}}{9 c^6}+\frac {100 \left (c^2 x^2-1\right )^{7/2}}{7 c^6}+\frac {6 \left (c^2 x^2-1\right )^{5/2}}{5 c^6}-\frac {8 \left (c^2 x^2-1\right )^{3/2}}{3 c^6}+\frac {16 \sqrt {c^2 x^2-1}}{c^6}\right )}{630 \sqrt {c x-1} \sqrt {c x+1}}\right )}{9 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 6298 |
| \(\displaystyle \frac {5}{9} d \left (\frac {3}{7} d \left (-\frac {2 b c \sqrt {d-c^2 d x^2} \left (\frac {1}{5} x^5 (a+b \text {arccosh}(c x))-\frac {1}{5} b c \int \frac {x^5}{\sqrt {c x-1} \sqrt {c x+1}}dx\right )}{5 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {d-c^2 d x^2} \int \frac {x^3 (a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{5 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{5} x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2\right )+\frac {1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{7} c^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{5} x^5 (a+b \text {arccosh}(c x))-\frac {1}{35} b c \left (\frac {19}{7} \left (\frac {x^4 \sqrt {c x-1} \sqrt {c x+1}}{5 c^2}+\frac {4 \left (\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^4}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}\right )}{5 c^2}\right )-\frac {5}{7} x^6 \sqrt {c x-1} \sqrt {c x+1}\right )\right )}{7 \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{9} x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{9} c^4 x^9 (a+b \text {arccosh}(c x))-\frac {2}{7} c^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{5} x^5 (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {70 \left (c^2 x^2-1\right )^{9/2}}{9 c^6}+\frac {100 \left (c^2 x^2-1\right )^{7/2}}{7 c^6}+\frac {6 \left (c^2 x^2-1\right )^{5/2}}{5 c^6}-\frac {8 \left (c^2 x^2-1\right )^{3/2}}{3 c^6}+\frac {16 \sqrt {c^2 x^2-1}}{c^6}\right )}{630 \sqrt {c x-1} \sqrt {c x+1}}\right )}{9 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 111 |
| \(\displaystyle \frac {5}{9} d \left (\frac {3}{7} d \left (-\frac {\sqrt {d-c^2 d x^2} \int \frac {x^3 (a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{5 \sqrt {c x-1} \sqrt {c x+1}}-\frac {2 b c \sqrt {d-c^2 d x^2} \left (\frac {1}{5} x^5 (a+b \text {arccosh}(c x))-\frac {1}{5} b c \left (\frac {\int \frac {4 x^3}{\sqrt {c x-1} \sqrt {c x+1}}dx}{5 c^2}+\frac {x^4 \sqrt {c x-1} \sqrt {c x+1}}{5 c^2}\right )\right )}{5 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{5} x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2\right )+\frac {1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{7} c^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{5} x^5 (a+b \text {arccosh}(c x))-\frac {1}{35} b c \left (\frac {19}{7} \left (\frac {x^4 \sqrt {c x-1} \sqrt {c x+1}}{5 c^2}+\frac {4 \left (\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^4}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}\right )}{5 c^2}\right )-\frac {5}{7} x^6 \sqrt {c x-1} \sqrt {c x+1}\right )\right )}{7 \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{9} x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{9} c^4 x^9 (a+b \text {arccosh}(c x))-\frac {2}{7} c^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{5} x^5 (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {70 \left (c^2 x^2-1\right )^{9/2}}{9 c^6}+\frac {100 \left (c^2 x^2-1\right )^{7/2}}{7 c^6}+\frac {6 \left (c^2 x^2-1\right )^{5/2}}{5 c^6}-\frac {8 \left (c^2 x^2-1\right )^{3/2}}{3 c^6}+\frac {16 \sqrt {c^2 x^2-1}}{c^6}\right )}{630 \sqrt {c x-1} \sqrt {c x+1}}\right )}{9 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{9} d \left (\frac {3}{7} d \left (-\frac {\sqrt {d-c^2 d x^2} \int \frac {x^3 (a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{5 \sqrt {c x-1} \sqrt {c x+1}}-\frac {2 b c \sqrt {d-c^2 d x^2} \left (\frac {1}{5} x^5 (a+b \text {arccosh}(c x))-\frac {1}{5} b c \left (\frac {4 \int \frac {x^3}{\sqrt {c x-1} \sqrt {c x+1}}dx}{5 c^2}+\frac {x^4 \sqrt {c x-1} \sqrt {c x+1}}{5 c^2}\right )\right )}{5 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{5} x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2\right )+\frac {1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{7} c^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{5} x^5 (a+b \text {arccosh}(c x))-\frac {1}{35} b c \left (\frac {19}{7} \left (\frac {x^4 \sqrt {c x-1} \sqrt {c x+1}}{5 c^2}+\frac {4 \left (\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^4}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}\right )}{5 c^2}\right )-\frac {5}{7} x^6 \sqrt {c x-1} \sqrt {c x+1}\right )\right )}{7 \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{9} x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{9} c^4 x^9 (a+b \text {arccosh}(c x))-\frac {2}{7} c^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{5} x^5 (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {70 \left (c^2 x^2-1\right )^{9/2}}{9 c^6}+\frac {100 \left (c^2 x^2-1\right )^{7/2}}{7 c^6}+\frac {6 \left (c^2 x^2-1\right )^{5/2}}{5 c^6}-\frac {8 \left (c^2 x^2-1\right )^{3/2}}{3 c^6}+\frac {16 \sqrt {c^2 x^2-1}}{c^6}\right )}{630 \sqrt {c x-1} \sqrt {c x+1}}\right )}{9 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 111 |
| \(\displaystyle \frac {5}{9} d \left (\frac {3}{7} d \left (-\frac {\sqrt {d-c^2 d x^2} \int \frac {x^3 (a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{5 \sqrt {c x-1} \sqrt {c x+1}}-\frac {2 b c \sqrt {d-c^2 d x^2} \left (\frac {1}{5} x^5 (a+b \text {arccosh}(c x))-\frac {1}{5} b c \left (\frac {4 \left (\frac {\int \frac {2 x}{\sqrt {c x-1} \sqrt {c x+1}}dx}{3 c^2}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}\right )}{5 c^2}+\frac {x^4 \sqrt {c x-1} \sqrt {c x+1}}{5 c^2}\right )\right )}{5 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{5} x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2\right )+\frac {1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{7} c^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{5} x^5 (a+b \text {arccosh}(c x))-\frac {1}{35} b c \left (\frac {19}{7} \left (\frac {x^4 \sqrt {c x-1} \sqrt {c x+1}}{5 c^2}+\frac {4 \left (\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^4}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}\right )}{5 c^2}\right )-\frac {5}{7} x^6 \sqrt {c x-1} \sqrt {c x+1}\right )\right )}{7 \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{9} x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{9} c^4 x^9 (a+b \text {arccosh}(c x))-\frac {2}{7} c^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{5} x^5 (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {70 \left (c^2 x^2-1\right )^{9/2}}{9 c^6}+\frac {100 \left (c^2 x^2-1\right )^{7/2}}{7 c^6}+\frac {6 \left (c^2 x^2-1\right )^{5/2}}{5 c^6}-\frac {8 \left (c^2 x^2-1\right )^{3/2}}{3 c^6}+\frac {16 \sqrt {c^2 x^2-1}}{c^6}\right )}{630 \sqrt {c x-1} \sqrt {c x+1}}\right )}{9 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{9} d \left (\frac {3}{7} d \left (-\frac {\sqrt {d-c^2 d x^2} \int \frac {x^3 (a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{5 \sqrt {c x-1} \sqrt {c x+1}}-\frac {2 b c \sqrt {d-c^2 d x^2} \left (\frac {1}{5} x^5 (a+b \text {arccosh}(c x))-\frac {1}{5} b c \left (\frac {4 \left (\frac {2 \int \frac {x}{\sqrt {c x-1} \sqrt {c x+1}}dx}{3 c^2}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}\right )}{5 c^2}+\frac {x^4 \sqrt {c x-1} \sqrt {c x+1}}{5 c^2}\right )\right )}{5 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{5} x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2\right )+\frac {1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{7} c^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{5} x^5 (a+b \text {arccosh}(c x))-\frac {1}{35} b c \left (\frac {19}{7} \left (\frac {x^4 \sqrt {c x-1} \sqrt {c x+1}}{5 c^2}+\frac {4 \left (\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^4}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}\right )}{5 c^2}\right )-\frac {5}{7} x^6 \sqrt {c x-1} \sqrt {c x+1}\right )\right )}{7 \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{9} x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{9} c^4 x^9 (a+b \text {arccosh}(c x))-\frac {2}{7} c^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{5} x^5 (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {70 \left (c^2 x^2-1\right )^{9/2}}{9 c^6}+\frac {100 \left (c^2 x^2-1\right )^{7/2}}{7 c^6}+\frac {6 \left (c^2 x^2-1\right )^{5/2}}{5 c^6}-\frac {8 \left (c^2 x^2-1\right )^{3/2}}{3 c^6}+\frac {16 \sqrt {c^2 x^2-1}}{c^6}\right )}{630 \sqrt {c x-1} \sqrt {c x+1}}\right )}{9 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 83 |
| \(\displaystyle \frac {5}{9} d \left (\frac {3}{7} d \left (-\frac {\sqrt {d-c^2 d x^2} \int \frac {x^3 (a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{5 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{5} x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {2 b c \sqrt {d-c^2 d x^2} \left (\frac {1}{5} x^5 (a+b \text {arccosh}(c x))-\frac {1}{5} b c \left (\frac {x^4 \sqrt {c x-1} \sqrt {c x+1}}{5 c^2}+\frac {4 \left (\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^4}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}\right )}{5 c^2}\right )\right )}{5 \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{7} c^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{5} x^5 (a+b \text {arccosh}(c x))-\frac {1}{35} b c \left (\frac {19}{7} \left (\frac {x^4 \sqrt {c x-1} \sqrt {c x+1}}{5 c^2}+\frac {4 \left (\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^4}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}\right )}{5 c^2}\right )-\frac {5}{7} x^6 \sqrt {c x-1} \sqrt {c x+1}\right )\right )}{7 \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{9} x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{9} c^4 x^9 (a+b \text {arccosh}(c x))-\frac {2}{7} c^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{5} x^5 (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {70 \left (c^2 x^2-1\right )^{9/2}}{9 c^6}+\frac {100 \left (c^2 x^2-1\right )^{7/2}}{7 c^6}+\frac {6 \left (c^2 x^2-1\right )^{5/2}}{5 c^6}-\frac {8 \left (c^2 x^2-1\right )^{3/2}}{3 c^6}+\frac {16 \sqrt {c^2 x^2-1}}{c^6}\right )}{630 \sqrt {c x-1} \sqrt {c x+1}}\right )}{9 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 6354 |
| \(\displaystyle \frac {1}{9} \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 x^4-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{9} c^4 (a+b \text {arccosh}(c x)) x^9-\frac {2}{7} c^2 (a+b \text {arccosh}(c x)) x^7+\frac {1}{5} (a+b \text {arccosh}(c x)) x^5-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {70 \left (c^2 x^2-1\right )^{9/2}}{9 c^6}+\frac {100 \left (c^2 x^2-1\right )^{7/2}}{7 c^6}+\frac {6 \left (c^2 x^2-1\right )^{5/2}}{5 c^6}-\frac {8 \left (c^2 x^2-1\right )^{3/2}}{3 c^6}+\frac {16 \sqrt {c^2 x^2-1}}{c^6}\right )}{630 \sqrt {c x-1} \sqrt {c x+1}}\right )}{9 \sqrt {c x-1} \sqrt {c x+1}}+\frac {5}{9} d \left (\frac {1}{7} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2 x^4-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{7} c^2 (a+b \text {arccosh}(c x)) x^7+\frac {1}{5} (a+b \text {arccosh}(c x)) x^5-\frac {1}{35} b c \left (\frac {19}{7} \left (\frac {\sqrt {c x-1} \sqrt {c x+1} x^4}{5 c^2}+\frac {4 \left (\frac {\sqrt {c x-1} \sqrt {c x+1} x^2}{3 c^2}+\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^4}\right )}{5 c^2}\right )-\frac {5}{7} x^6 \sqrt {c x-1} \sqrt {c x+1}\right )\right )}{7 \sqrt {c x-1} \sqrt {c x+1}}+\frac {3}{7} d \left (\frac {1}{5} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 x^4-\frac {2 b c \sqrt {d-c^2 d x^2} \left (\frac {1}{5} x^5 (a+b \text {arccosh}(c x))-\frac {1}{5} b c \left (\frac {\sqrt {c x-1} \sqrt {c x+1} x^4}{5 c^2}+\frac {4 \left (\frac {\sqrt {c x-1} \sqrt {c x+1} x^2}{3 c^2}+\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^4}\right )}{5 c^2}\right )\right )}{5 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {d-c^2 d x^2} \left (\frac {x^2 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^2}{3 c^2}-\frac {2 b \int x^2 (a+b \text {arccosh}(c x))dx}{3 c}+\frac {2 \int \frac {x (a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{3 c^2}\right )}{5 \sqrt {c x-1} \sqrt {c x+1}}\right )\right )\) |
| \(\Big \downarrow \) 6298 |
| \(\displaystyle \frac {1}{9} \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 x^4-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{9} c^4 (a+b \text {arccosh}(c x)) x^9-\frac {2}{7} c^2 (a+b \text {arccosh}(c x)) x^7+\frac {1}{5} (a+b \text {arccosh}(c x)) x^5-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {70 \left (c^2 x^2-1\right )^{9/2}}{9 c^6}+\frac {100 \left (c^2 x^2-1\right )^{7/2}}{7 c^6}+\frac {6 \left (c^2 x^2-1\right )^{5/2}}{5 c^6}-\frac {8 \left (c^2 x^2-1\right )^{3/2}}{3 c^6}+\frac {16 \sqrt {c^2 x^2-1}}{c^6}\right )}{630 \sqrt {c x-1} \sqrt {c x+1}}\right )}{9 \sqrt {c x-1} \sqrt {c x+1}}+\frac {5}{9} d \left (\frac {1}{7} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2 x^4-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{7} c^2 (a+b \text {arccosh}(c x)) x^7+\frac {1}{5} (a+b \text {arccosh}(c x)) x^5-\frac {1}{35} b c \left (\frac {19}{7} \left (\frac {\sqrt {c x-1} \sqrt {c x+1} x^4}{5 c^2}+\frac {4 \left (\frac {\sqrt {c x-1} \sqrt {c x+1} x^2}{3 c^2}+\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^4}\right )}{5 c^2}\right )-\frac {5}{7} x^6 \sqrt {c x-1} \sqrt {c x+1}\right )\right )}{7 \sqrt {c x-1} \sqrt {c x+1}}+\frac {3}{7} d \left (\frac {1}{5} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 x^4-\frac {2 b c \sqrt {d-c^2 d x^2} \left (\frac {1}{5} x^5 (a+b \text {arccosh}(c x))-\frac {1}{5} b c \left (\frac {\sqrt {c x-1} \sqrt {c x+1} x^4}{5 c^2}+\frac {4 \left (\frac {\sqrt {c x-1} \sqrt {c x+1} x^2}{3 c^2}+\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^4}\right )}{5 c^2}\right )\right )}{5 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {d-c^2 d x^2} \left (\frac {x^2 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^2}{3 c^2}-\frac {2 b \left (\frac {1}{3} x^3 (a+b \text {arccosh}(c x))-\frac {1}{3} b c \int \frac {x^3}{\sqrt {c x-1} \sqrt {c x+1}}dx\right )}{3 c}+\frac {2 \int \frac {x (a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{3 c^2}\right )}{5 \sqrt {c x-1} \sqrt {c x+1}}\right )\right )\) |
| \(\Big \downarrow \) 111 |
| \(\displaystyle \frac {1}{9} \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 x^4-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{9} c^4 (a+b \text {arccosh}(c x)) x^9-\frac {2}{7} c^2 (a+b \text {arccosh}(c x)) x^7+\frac {1}{5} (a+b \text {arccosh}(c x)) x^5-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {70 \left (c^2 x^2-1\right )^{9/2}}{9 c^6}+\frac {100 \left (c^2 x^2-1\right )^{7/2}}{7 c^6}+\frac {6 \left (c^2 x^2-1\right )^{5/2}}{5 c^6}-\frac {8 \left (c^2 x^2-1\right )^{3/2}}{3 c^6}+\frac {16 \sqrt {c^2 x^2-1}}{c^6}\right )}{630 \sqrt {c x-1} \sqrt {c x+1}}\right )}{9 \sqrt {c x-1} \sqrt {c x+1}}+\frac {5}{9} d \left (\frac {1}{7} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2 x^4-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{7} c^2 (a+b \text {arccosh}(c x)) x^7+\frac {1}{5} (a+b \text {arccosh}(c x)) x^5-\frac {1}{35} b c \left (\frac {19}{7} \left (\frac {\sqrt {c x-1} \sqrt {c x+1} x^4}{5 c^2}+\frac {4 \left (\frac {\sqrt {c x-1} \sqrt {c x+1} x^2}{3 c^2}+\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^4}\right )}{5 c^2}\right )-\frac {5}{7} x^6 \sqrt {c x-1} \sqrt {c x+1}\right )\right )}{7 \sqrt {c x-1} \sqrt {c x+1}}+\frac {3}{7} d \left (\frac {1}{5} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 x^4-\frac {2 b c \sqrt {d-c^2 d x^2} \left (\frac {1}{5} x^5 (a+b \text {arccosh}(c x))-\frac {1}{5} b c \left (\frac {\sqrt {c x-1} \sqrt {c x+1} x^4}{5 c^2}+\frac {4 \left (\frac {\sqrt {c x-1} \sqrt {c x+1} x^2}{3 c^2}+\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^4}\right )}{5 c^2}\right )\right )}{5 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {d-c^2 d x^2} \left (\frac {x^2 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^2}{3 c^2}-\frac {2 b \left (\frac {1}{3} x^3 (a+b \text {arccosh}(c x))-\frac {1}{3} b c \left (\frac {\sqrt {c x-1} \sqrt {c x+1} x^2}{3 c^2}+\frac {\int \frac {2 x}{\sqrt {c x-1} \sqrt {c x+1}}dx}{3 c^2}\right )\right )}{3 c}+\frac {2 \int \frac {x (a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{3 c^2}\right )}{5 \sqrt {c x-1} \sqrt {c x+1}}\right )\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 x^4-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{9} c^4 (a+b \text {arccosh}(c x)) x^9-\frac {2}{7} c^2 (a+b \text {arccosh}(c x)) x^7+\frac {1}{5} (a+b \text {arccosh}(c x)) x^5-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {70 \left (c^2 x^2-1\right )^{9/2}}{9 c^6}+\frac {100 \left (c^2 x^2-1\right )^{7/2}}{7 c^6}+\frac {6 \left (c^2 x^2-1\right )^{5/2}}{5 c^6}-\frac {8 \left (c^2 x^2-1\right )^{3/2}}{3 c^6}+\frac {16 \sqrt {c^2 x^2-1}}{c^6}\right )}{630 \sqrt {c x-1} \sqrt {c x+1}}\right )}{9 \sqrt {c x-1} \sqrt {c x+1}}+\frac {5}{9} d \left (\frac {1}{7} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2 x^4-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{7} c^2 (a+b \text {arccosh}(c x)) x^7+\frac {1}{5} (a+b \text {arccosh}(c x)) x^5-\frac {1}{35} b c \left (\frac {19}{7} \left (\frac {\sqrt {c x-1} \sqrt {c x+1} x^4}{5 c^2}+\frac {4 \left (\frac {\sqrt {c x-1} \sqrt {c x+1} x^2}{3 c^2}+\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^4}\right )}{5 c^2}\right )-\frac {5}{7} x^6 \sqrt {c x-1} \sqrt {c x+1}\right )\right )}{7 \sqrt {c x-1} \sqrt {c x+1}}+\frac {3}{7} d \left (\frac {1}{5} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 x^4-\frac {2 b c \sqrt {d-c^2 d x^2} \left (\frac {1}{5} x^5 (a+b \text {arccosh}(c x))-\frac {1}{5} b c \left (\frac {\sqrt {c x-1} \sqrt {c x+1} x^4}{5 c^2}+\frac {4 \left (\frac {\sqrt {c x-1} \sqrt {c x+1} x^2}{3 c^2}+\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^4}\right )}{5 c^2}\right )\right )}{5 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {d-c^2 d x^2} \left (\frac {x^2 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^2}{3 c^2}-\frac {2 b \left (\frac {1}{3} x^3 (a+b \text {arccosh}(c x))-\frac {1}{3} b c \left (\frac {\sqrt {c x-1} \sqrt {c x+1} x^2}{3 c^2}+\frac {2 \int \frac {x}{\sqrt {c x-1} \sqrt {c x+1}}dx}{3 c^2}\right )\right )}{3 c}+\frac {2 \int \frac {x (a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{3 c^2}\right )}{5 \sqrt {c x-1} \sqrt {c x+1}}\right )\right )\) |
3.2.86.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] && EqQ[a*d*f *(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)), 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*(a + b*x)^(m - 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/(d*f*(m + n + p + 1))), x] + Simp[1/(d*f*(m + n + p + 1)) Int[(a + b*x) ^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m - 1) + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] & & GtQ[m, 1] && NeQ[m + n + p + 1, 0] && IntegerQ[m]
Int[((e_.)*(x_))^(m_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p_.)*((a2_) + (b2_.) *(x_)^(non2_.))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[d*(e*x)^( m + 1)*(a1 + b1*x^(n/2))^(p + 1)*((a2 + b2*x^(n/2))^(p + 1)/(b1*b2*e*(m + n *(p + 1) + 1))), x] - Simp[(a1*a2*d*(m + 1) - b1*b2*c*(m + n*(p + 1) + 1))/ (b1*b2*(m + n*(p + 1) + 1)) Int[(e*x)^m*(a1 + b1*x^(n/2))^p*(a2 + b2*x^(n /2))^p, x], x] /; FreeQ[{a1, b1, a2, b2, c, d, e, m, n, p}, x] && EqQ[non2, n/2] && EqQ[a2*b1 + a1*b2, 0] && NeQ[m + n*(p + 1) + 1, 0]
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x _) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x ] && IGtQ[p, 0]
Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_ )^4)^(p_.), x_Symbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && Int egerQ[(m - 1)/2]
Int[((f_.)*(x_))^(m_.)*((d1_) + (e1_.)*(x_)^(non2_.))^(q_.)*((d2_) + (e2_.) *(x_)^(non2_.))^(q_.)*((a_.) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_))^(p_.), x _Symbol] :> Simp[(d1 + e1*x^(n/2))^FracPart[q]*((d2 + e2*x^(n/2))^FracPart[ q]/(d1*d2 + e1*e2*x^n)^FracPart[q]) Int[(f*x)^m*(d1*d2 + e1*e2*x^n)^q*(a + b*x^n + c*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[non2, n/2] && EqQ[d2*e1 + d1*e2, 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcCosh[c*x])^n/(d*(m + 1))), x] - Simp[b*c* (n/(d*(m + 1))) Int[(d*x)^(m + 1)*((a + b*ArcCosh[c*x])^(n - 1)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] & & NeQ[m, -1]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_.)*((d1_) + ( e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_.), x_Symbol] :> Int[(f*x)^m*(d1 *d2 + e1*e2*x^2)^p*(a + b*ArcCosh[c*x])^n, x] /; FreeQ[{a, b, c, d1, e1, d2 , e2, f, m, n}, x] && EqQ[d2*e1 + d1*e2, 0] && IntegerQ[p]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_ )^2)^(p_.), x_Symbol] :> With[{u = IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Simp [(a + b*ArcCosh[c*x]) u, x] - Simp[b*c Int[SimplifyIntegrand[u/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x], x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && E qQ[c^2*d + e, 0] && IGtQ[p, 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(f*x)^(m + 1)*Sqrt[d + e*x^2]*((a + b*Arc Cosh[c*x])^n/(f*(m + 2))), x] + (-Simp[(1/(m + 2))*Simp[Sqrt[d + e*x^2]/(Sq rt[1 + c*x]*Sqrt[-1 + c*x])] Int[(f*x)^m*((a + b*ArcCosh[c*x])^n/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])), x], x] - Simp[b*c*(n/(f*(m + 2)))*Simp[Sqrt[d + e* x^2]/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])] Int[(f*x)^(m + 1)*(a + b*ArcCosh[c*x ])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0] && (IGtQ[m, -2] || EqQ[n, 1])
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_ .)*(x_)^2)^(p_.), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^p*((a + b*Arc Cosh[c*x])^n/(f*(m + 2*p + 1))), x] + (Simp[2*d*(p/(m + 2*p + 1)) Int[(f* x)^m*(d + e*x^2)^(p - 1)*(a + b*ArcCosh[c*x])^n, x], x] - Simp[b*c*(n/(f*(m + 2*p + 1)))*Simp[(d + e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)] Int[(f*x)^(m + 1)*(1 + c*x)^(p - 1/2)*(-1 + c*x)^(p - 1/2)*(a + b*ArcCosh[c*x])^(n - 1) , x], x]) /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0] && !LtQ[m, -1]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d1_) + (e 1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_))^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*((a + b*ArcCosh[c*x])^n/(e1*e2*( m + 2*p + 1))), x] + (Simp[f^2*((m - 1)/(c^2*(m + 2*p + 1))) Int[(f*x)^(m - 2)*(d1 + e1*x)^p*(d2 + e2*x)^p*(a + b*ArcCosh[c*x])^n, x], x] - Simp[b*f *(n/(c*(m + 2*p + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/( -1 + c*x)^p] Int[(f*x)^(m - 1)*(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*( a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d1, e1, d2, e2, f, p}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && GtQ[n, 0] && IGtQ[m, 1] && N eQ[m + 2*p + 1, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(2223\) vs. \(2(776)=1552\).
Time = 0.78 (sec) , antiderivative size = 2224, normalized size of antiderivative = 2.53
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(2224\) |
| parts | \(\text {Expression too large to display}\) | \(2224\) |
a^2*(-1/9*x^2*(-c^2*d*x^2+d)^(7/2)/c^2/d-2/63/d/c^4*(-c^2*d*x^2+d)^(7/2))+ b^2*(1/373248*(-d*(c^2*x^2-1))^(1/2)*(256*c^10*x^10-704*c^8*x^8+256*(c*x+1 )^(1/2)*(c*x-1)^(1/2)*x^9*c^9+688*c^6*x^6-576*(c*x+1)^(1/2)*(c*x-1)^(1/2)* x^7*c^7-280*c^4*x^4+432*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^5*x^5+41*c^2*x^2-120 *(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^3*x^3+9*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x-1)* (81*arccosh(c*x)^2-18*arccosh(c*x)+2)*d^2/(c*x+1)/c^4/(c*x-1)-3/175616*(-d *(c^2*x^2-1))^(1/2)*(64*c^8*x^8-144*c^6*x^6+64*(c*x+1)^(1/2)*(c*x-1)^(1/2) *x^7*c^7+104*c^4*x^4-112*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^5*x^5-25*c^2*x^2+56 *(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^3*x^3-7*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x+1)* (49*arccosh(c*x)^2-14*arccosh(c*x)+2)*d^2/(c*x+1)/c^4/(c*x-1)+1/1728*(-d*( c^2*x^2-1))^(1/2)*(4*c^4*x^4-5*c^2*x^2+4*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^3*x ^3-3*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x+1)*(9*arccosh(c*x)^2-6*arccosh(c*x)+2 )*d^2/(c*x+1)/c^4/(c*x-1)-3/256*(-d*(c^2*x^2-1))^(1/2)*((c*x-1)^(1/2)*(c*x +1)^(1/2)*c*x+c^2*x^2-1)*(arccosh(c*x)^2-2*arccosh(c*x)+2)*d^2/(c*x+1)/c^4 /(c*x-1)-3/256*(-d*(c^2*x^2-1))^(1/2)*(-(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x+c^ 2*x^2-1)*(arccosh(c*x)^2+2*arccosh(c*x)+2)*d^2/(c*x+1)/c^4/(c*x-1)+1/1728* (-d*(c^2*x^2-1))^(1/2)*(-4*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^3*x^3+4*c^4*x^4+3 *(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x-5*c^2*x^2+1)*(9*arccosh(c*x)^2+6*arccosh( c*x)+2)*d^2/(c*x+1)/c^4/(c*x-1)-3/175616*(-d*(c^2*x^2-1))^(1/2)*(-64*(c*x+ 1)^(1/2)*(c*x-1)^(1/2)*x^7*c^7+64*c^8*x^8+112*(c*x+1)^(1/2)*(c*x-1)^(1/...
Time = 0.28 (sec) , antiderivative size = 558, normalized size of antiderivative = 0.63 \[ \int x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 \, dx=\frac {3969 \, {\left (7 \, b^{2} c^{10} d^{2} x^{10} - 26 \, b^{2} c^{8} d^{2} x^{8} + 34 \, b^{2} c^{6} d^{2} x^{6} - 16 \, b^{2} c^{4} d^{2} x^{4} - b^{2} c^{2} d^{2} x^{2} + 2 \, b^{2} d^{2}\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right )^{2} - 126 \, {\left (49 \, a b c^{9} d^{2} x^{9} - 171 \, a b c^{7} d^{2} x^{7} + 189 \, a b c^{5} d^{2} x^{5} - 21 \, a b c^{3} d^{2} x^{3} - 126 \, a b c d^{2} x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} - 126 \, {\left ({\left (49 \, b^{2} c^{9} d^{2} x^{9} - 171 \, b^{2} c^{7} d^{2} x^{7} + 189 \, b^{2} c^{5} d^{2} x^{5} - 21 \, b^{2} c^{3} d^{2} x^{3} - 126 \, b^{2} c d^{2} x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} - 63 \, {\left (7 \, a b c^{10} d^{2} x^{10} - 26 \, a b c^{8} d^{2} x^{8} + 34 \, a b c^{6} d^{2} x^{6} - 16 \, a b c^{4} d^{2} x^{4} - a b c^{2} d^{2} x^{2} + 2 \, a b d^{2}\right )} \sqrt {-c^{2} d x^{2} + d}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + {\left (343 \, {\left (81 \, a^{2} + 2 \, b^{2}\right )} c^{10} d^{2} x^{10} - 2 \, {\left (51597 \, a^{2} + 1490 \, b^{2}\right )} c^{8} d^{2} x^{8} + 2 \, {\left (67473 \, a^{2} + 2152 \, b^{2}\right )} c^{6} d^{2} x^{6} - 4 \, {\left (15876 \, a^{2} + 53 \, b^{2}\right )} c^{4} d^{2} x^{4} - {\left (3969 \, a^{2} + 14078 \, b^{2}\right )} c^{2} d^{2} x^{2} + 2 \, {\left (3969 \, a^{2} + 6140 \, b^{2}\right )} d^{2}\right )} \sqrt {-c^{2} d x^{2} + d}}{250047 \, {\left (c^{6} x^{2} - c^{4}\right )}} \]
1/250047*(3969*(7*b^2*c^10*d^2*x^10 - 26*b^2*c^8*d^2*x^8 + 34*b^2*c^6*d^2* x^6 - 16*b^2*c^4*d^2*x^4 - b^2*c^2*d^2*x^2 + 2*b^2*d^2)*sqrt(-c^2*d*x^2 + d)*log(c*x + sqrt(c^2*x^2 - 1))^2 - 126*(49*a*b*c^9*d^2*x^9 - 171*a*b*c^7* d^2*x^7 + 189*a*b*c^5*d^2*x^5 - 21*a*b*c^3*d^2*x^3 - 126*a*b*c*d^2*x)*sqrt (-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1) - 126*((49*b^2*c^9*d^2*x^9 - 171*b^2*c^ 7*d^2*x^7 + 189*b^2*c^5*d^2*x^5 - 21*b^2*c^3*d^2*x^3 - 126*b^2*c*d^2*x)*sq rt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1) - 63*(7*a*b*c^10*d^2*x^10 - 26*a*b*c^ 8*d^2*x^8 + 34*a*b*c^6*d^2*x^6 - 16*a*b*c^4*d^2*x^4 - a*b*c^2*d^2*x^2 + 2* a*b*d^2)*sqrt(-c^2*d*x^2 + d))*log(c*x + sqrt(c^2*x^2 - 1)) + (343*(81*a^2 + 2*b^2)*c^10*d^2*x^10 - 2*(51597*a^2 + 1490*b^2)*c^8*d^2*x^8 + 2*(67473* a^2 + 2152*b^2)*c^6*d^2*x^6 - 4*(15876*a^2 + 53*b^2)*c^4*d^2*x^4 - (3969*a ^2 + 14078*b^2)*c^2*d^2*x^2 + 2*(3969*a^2 + 6140*b^2)*d^2)*sqrt(-c^2*d*x^2 + d))/(c^6*x^2 - c^4)
Timed out. \[ \int x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 \, dx=\text {Timed out} \]
Time = 0.31 (sec) , antiderivative size = 471, normalized size of antiderivative = 0.54 \[ \int x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 \, dx=-\frac {1}{63} \, {\left (\frac {7 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} x^{2}}{c^{2} d} + \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}}}{c^{4} d}\right )} b^{2} \operatorname {arcosh}\left (c x\right )^{2} - \frac {2}{63} \, {\left (\frac {7 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} x^{2}}{c^{2} d} + \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}}}{c^{4} d}\right )} a b \operatorname {arcosh}\left (c x\right ) - \frac {1}{63} \, {\left (\frac {7 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} x^{2}}{c^{2} d} + \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}}}{c^{4} d}\right )} a^{2} + \frac {2}{250047} \, b^{2} {\left (\frac {343 \, \sqrt {c^{2} x^{2} - 1} c^{6} \sqrt {-d} d^{2} x^{8} - 1147 \, \sqrt {c^{2} x^{2} - 1} c^{4} \sqrt {-d} d^{2} x^{6} + 1005 \, \sqrt {c^{2} x^{2} - 1} c^{2} \sqrt {-d} d^{2} x^{4} + 899 \, \sqrt {c^{2} x^{2} - 1} \sqrt {-d} d^{2} x^{2} - \frac {6140 \, \sqrt {c^{2} x^{2} - 1} \sqrt {-d} d^{2}}{c^{2}}}{c^{2}} - \frac {63 \, {\left (49 \, c^{8} \sqrt {-d} d^{2} x^{9} - 171 \, c^{6} \sqrt {-d} d^{2} x^{7} + 189 \, c^{4} \sqrt {-d} d^{2} x^{5} - 21 \, c^{2} \sqrt {-d} d^{2} x^{3} - 126 \, \sqrt {-d} d^{2} x\right )} \operatorname {arcosh}\left (c x\right )}{c^{3}}\right )} - \frac {2 \, {\left (49 \, c^{8} \sqrt {-d} d^{2} x^{9} - 171 \, c^{6} \sqrt {-d} d^{2} x^{7} + 189 \, c^{4} \sqrt {-d} d^{2} x^{5} - 21 \, c^{2} \sqrt {-d} d^{2} x^{3} - 126 \, \sqrt {-d} d^{2} x\right )} a b}{3969 \, c^{3}} \]
-1/63*(7*(-c^2*d*x^2 + d)^(7/2)*x^2/(c^2*d) + 2*(-c^2*d*x^2 + d)^(7/2)/(c^ 4*d))*b^2*arccosh(c*x)^2 - 2/63*(7*(-c^2*d*x^2 + d)^(7/2)*x^2/(c^2*d) + 2* (-c^2*d*x^2 + d)^(7/2)/(c^4*d))*a*b*arccosh(c*x) - 1/63*(7*(-c^2*d*x^2 + d )^(7/2)*x^2/(c^2*d) + 2*(-c^2*d*x^2 + d)^(7/2)/(c^4*d))*a^2 + 2/250047*b^2 *((343*sqrt(c^2*x^2 - 1)*c^6*sqrt(-d)*d^2*x^8 - 1147*sqrt(c^2*x^2 - 1)*c^4 *sqrt(-d)*d^2*x^6 + 1005*sqrt(c^2*x^2 - 1)*c^2*sqrt(-d)*d^2*x^4 + 899*sqrt (c^2*x^2 - 1)*sqrt(-d)*d^2*x^2 - 6140*sqrt(c^2*x^2 - 1)*sqrt(-d)*d^2/c^2)/ c^2 - 63*(49*c^8*sqrt(-d)*d^2*x^9 - 171*c^6*sqrt(-d)*d^2*x^7 + 189*c^4*sqr t(-d)*d^2*x^5 - 21*c^2*sqrt(-d)*d^2*x^3 - 126*sqrt(-d)*d^2*x)*arccosh(c*x) /c^3) - 2/3969*(49*c^8*sqrt(-d)*d^2*x^9 - 171*c^6*sqrt(-d)*d^2*x^7 + 189*c ^4*sqrt(-d)*d^2*x^5 - 21*c^2*sqrt(-d)*d^2*x^3 - 126*sqrt(-d)*d^2*x)*a*b/c^ 3
Exception generated. \[ \int x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 \, dx=\int x^3\,{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^{5/2} \,d x \]